{"id":12725,"date":"2021-03-04T10:37:59","date_gmt":"2021-03-04T05:07:59","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=12725"},"modified":"2022-03-28T20:03:35","modified_gmt":"2022-03-28T14:33:35","slug":"2-3-application-of-scalar-product","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=12725","title":{"rendered":"2.3    APPLICATION OF SCALAR AND   VECTOR PRODUCT"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">Application of Scalar Product<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Work done<\/h4>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img decoding=\"async\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcRxMqXSprcGu0vFEQizyOo-SvB2pOkSeYHAIA&amp;usqp=CAU\" alt=\"Multiplying vectors together | Work physics, Physics, Angles math\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\ done = \\overrightarrow{F}.\\overrightarrow{d}\\ where\\ \\overrightarrow{d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- sidebar ad 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:326px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6703350399\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p class=\"has-text-align-center\"><strong>PART &#8211; B<\/strong><\/p>\n\n\n\n<p> <strong>Example &nbsp;:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Find\\ the\\ work\\ done\\ by\\ the\\ force\\  3\\overrightarrow{i}+ 5\\overrightarrow{j}+ 7\\overrightarrow{k},\\ when\\ the\\ displacement\\ is\\  2\\overrightarrow{i}- \\overrightarrow{j} +\\overrightarrow{k}\\]<\/div>\n\n\n\n<p> Soln:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 3\\overrightarrow{i}+ 5\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}= 2\\overrightarrow{i}- \\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (3\\overrightarrow{i}+ 5\\overrightarrow{j}+ 7\\overrightarrow{k}) .(2\\overrightarrow{i}- \\overrightarrow{j}+ \\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> =   3 ( 2 ) + 5 ( &#8211; 1 ) + 7 ( 1 ) <\/p>\n\n\n\n<p class=\"has-text-align-center\"> =&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 5&nbsp; +&nbsp; 7<\/p>\n\n\n\n<p class=\"has-text-align-center\"> Work done  = 8 units<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p><strong>Example &nbsp;:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ particle\\ acted\\ on\\  by\\ the\\ forces\\ 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} and\\  \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} acting\\ on\\ the\\ particle\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[displaces\\ the\\ particle\\ from\\ the\\  point\\  \\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k} to\\  the\\ point\\  3\\overrightarrow{i}- 5\\overrightarrow{j}+4\\overrightarrow{k}.  Find\\ the\\ total\\ work\\ done\\ by\\ the\\ forces.\\]<\/div>\n\n\n\n<p>Soln:      <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_1}= 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} = 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} + \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 4\\overrightarrow{i}+ 9\\overrightarrow{j}+4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= \\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 3\\overrightarrow{i}-5\\overrightarrow{j}+ 4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow {d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j} + 4\\overrightarrow{k}- (\\overrightarrow{i}\\ + 2\\overrightarrow{j}+3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i}\\ &#8211; \\overrightarrow{j} + 3\\overrightarrow{k}- \\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}= 2\\overrightarrow{i}- 7\\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (4\\overrightarrow{i}+ 9\\overrightarrow{j}+ 4\\overrightarrow{k}) .(2\\overrightarrow{i}-7 \\overrightarrow{j}+ 1\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=   4 ( 2 ) + 9 ( &#8211; 7 ) + 4 ( 1 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 8&nbsp; &#8211; 63&nbsp; +&nbsp; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;&nbsp;&nbsp; Work done&nbsp; =&nbsp; &#8211;51 units<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done&nbsp; =&nbsp; 51 units&nbsp; (by taking positive value)<\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<!-- admitad.banner: 3785j823mq109003a68816525dc3e8 AliExpress WW -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/alitems.com\/g\/3785j823mq109003a68816525dc3e8\/?i=4\"><img decoding=\"async\" width=\"125\" height=\"125\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/3785j823mq109003a68816525dc3e8\/\" alt=\"AliExpress WW\"><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<!-- admitad.banner: 42vtejotyw109003a688c26e6ba059 Decathlon [CPS] IN -->\n<a target=\"_blank\" rel=\"nofollow noopener\" href=\"https:\/\/ad.admitad.com\/g\/42vtejotyw109003a688c26e6ba059\/?i=4&amp;subid=https%3A%2F%2Fad.admitad.com%2Fg%2F0pgslvb2vv109003a688c26e6ba059%2F%3Fsubid%3Dhttps%253A%252F%252Fad.admitad.com%252Fg%252F0pgslvb2vv109003a688c26e6ba059%252F\"><img decoding=\"async\" width=\"200\" height=\"200\" border=\"0\" src=\"https:\/\/ad.admitad.com\/b\/42vtejotyw109003a688c26e6ba059\/\" alt=\"Decathlon [CPS] IN\"><\/a>\n<!-- \/admitad.banner -->\n\n\n\n<p><strong>Example &nbsp;:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ particle\\ acted\\ on\\  by\\ the\\ forces\\ 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k} and\\  2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\]<\/div>\n\n\n\n<p>                      is displaced from the point ( 1, 1,&nbsp; 1 )&nbsp; to the&nbsp; point&nbsp; ( 2, &#8211; 3, 5 ).&nbsp; Find the total work done.<\/p>\n\n\n\n<p>Soln:        <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_1}= 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= 2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} = 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k} + 2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 6\\overrightarrow{i}+ 10\\overrightarrow{j}-\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= \\overrightarrow{i}+ \\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}-3\\overrightarrow{j}+ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow {d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}- (\\overrightarrow{i}\\ + \\overrightarrow{j}+\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}- \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}= \\overrightarrow{i}- 4\\overrightarrow{j}+4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (6\\overrightarrow{i}+ 10\\overrightarrow{j}-\\overrightarrow{k}) .(\\overrightarrow{i}- 4\\overrightarrow{j}+4\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;&nbsp;&nbsp; =&nbsp; 6 ( 1 )&nbsp; + 10 ( -4 ) &#8211; 1 ( 4 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 40&nbsp; &#8211;&nbsp; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; &#8211; 38<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done&nbsp; &nbsp;&nbsp;= &nbsp;&#8211;38 units<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done&nbsp; &nbsp;&nbsp;=&nbsp; 38 units&nbsp; (by taking positive value)<\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6t+ed+2i-1n-4w\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9770958327\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p> <strong>Example &nbsp;:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ particle\\ is\\ displaced\\  from\\ the\\ point\\ 5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k}\\ to\\  the\\ point\\ 6\\overrightarrow{i}+ 2\\overrightarrow{j}-2\\overrightarrow{k}\\]\\[under\\ the\\ action\\ of\\ forces\\ 10\\overrightarrow{i}- \\overrightarrow{j} + 11\\overrightarrow{k},\\ 4\\overrightarrow{i} + 5\\overrightarrow{j} +6\\overrightarrow{k}\\ and\\ -2\\overrightarrow{i} + \\overrightarrow{j}- 9\\overrightarrow{k}.\\]\\[Calculate\\ the\\ total\\ work\\ done\\ by\\ the\\ forces.\\]<\/div>\n\n\n\n<p> Soln:  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_1}= 10\\overrightarrow{i}- \\overrightarrow{j}+ 11\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= 4\\overrightarrow{i} + 5\\overrightarrow{j}+ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_3}= -2\\overrightarrow{i}+ \\overrightarrow{j}- 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} + \\overrightarrow{F_3}\\]\\[= 10\\overrightarrow{i}- \\overrightarrow{j}+ 11\\overrightarrow{k} + 4\\overrightarrow{i} + 5\\overrightarrow{j}+ 6\\overrightarrow{k}- 2\\overrightarrow{i} + \\overrightarrow{j}- 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 12\\overrightarrow{i}+ 5\\overrightarrow{j} +8\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 6\\overrightarrow{i} +2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow {d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i} +2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}- (5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i} +2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}- 5\\overrightarrow{i} + 5\\overrightarrow{j}+ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}= \\overrightarrow{i}+ 7\\overrightarrow{j}+5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (12\\overrightarrow{i}+ 5\\overrightarrow{j} +8\\overrightarrow{k}) .(\\overrightarrow{i}+ 7\\overrightarrow{j}+5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=12(1) + 5(7) + 8(5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 12 + 35 + 40\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\ done\\ = 87\\ units\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/shareasale.com\/r.cfm?b=1787431&amp;u=2863840&amp;m=73004&amp;urllink=&amp;afftrack=\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/static.shareasale.com\/image\/73004\/2005702_300x250.jpg?ssl=1\" border=\"0\" alt=\"Thanos (Modern Version) Statue by Sideshow Collectibles\"><\/a>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/shareasale.com\/r.cfm?b=1724747&amp;u=2863840&amp;m=16058&amp;urllink=&amp;afftrack=\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/static.shareasale.com\/image\/16058\/300x250_07_00.gif?ssl=1\" border=\"0\" alt=\"New 3D Puzzle\"><\/a>\n\n\n\n<h3 class=\"wp-block-heading\">Application of Vector Product<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Moment (or) Torque of a force about a point<\/strong><\/h4>\n\n\n\n<figure class=\"wp-block-image aligncenter is-resized\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/qsstudy.com\/wp-content\/uploads\/2017\/06\/Torque.jpg?resize=309%2C161&#038;ssl=1\" alt=\"Torque or Moment of a Force - QS Study\" width=\"309\" height=\"161\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A\\ be\\ any\\ point\\ and\\ \\overrightarrow{r}\\ be\\ the\\ position\\ vector\\ relative\\ to\\ the\\ point\\ A\\]\\[of\\ any\\ point\\ P\\ on\\ the\\ line\\ of\\ the\\ action\\ of\\ the\\ force\\ \\overrightarrow{F}.\\]\\[The\\ moment\\ of\\ the\\ force\\ about\\ the\\ point\\ O\\ is\\ defined\\ as\\ \\overrightarrow{M}= \\overrightarrow{r}\u00d7 \\overrightarrow{F}\\]\\[where\\  \\overrightarrow{r}= \\overrightarrow{AP}= \\overrightarrow{OP}- \\overrightarrow{OA}\\].<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ The\\ Magnitude \\ of\\ Moment = |\\overrightarrow{r} \u00d7 \\overrightarrow{F}|\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3040040190\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p><strong>Example &nbsp;:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Find\\ the\\ moment\\ of\\ the\\ force\\ 3\\overrightarrow{i}+\\overrightarrow{k}\\ acting\\ through\\ the\\ point\\ \\overrightarrow{i}+2\\overrightarrow{j}-\\overrightarrow{k} about\\ the\\ point\\  2\\overrightarrow{i}+ \\overrightarrow{j}-2\\overrightarrow{k}.\\]<\/div>\n\n\n\n<p>Soln:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 3\\overrightarrow{i} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= \\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 2\\overrightarrow{i} +\\overrightarrow{j} -2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}= \\overrightarrow{AP} = \\overrightarrow{OP}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}- (2\\overrightarrow{i} +\\overrightarrow{j} -2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}- 2\\overrightarrow{i}-\\overrightarrow{j} +2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}=  -\\overrightarrow{i} + \\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Moment = \\overrightarrow{r}\u00d7 \\overrightarrow{F}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp;\\overrightarrow{k}\\\\\n-1 &amp; 1 &amp; 1\\\\\n3 &amp; 0 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 &#8211; 0) -\\overrightarrow{j}(-1 &#8211; 3)+\\overrightarrow{k}(0 &#8211; 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(1) -\\overrightarrow{j}(-4)+\\overrightarrow{k}(-3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}\u00d7 \\overrightarrow{F}= \\overrightarrow{i}+ 4\\overrightarrow{j}-3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Magnitude\\ of \\ Moment = |\\overrightarrow{r} \u00d7 \\overrightarrow{F}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\sqrt{(1)^2 + (4)^2 + (-3)^2 }=\\sqrt{(1 + 16 +9 }=\\sqrt{26}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Magnitude\\ of \\ Moment = \\sqrt{26}\\ units\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p><strong>Example &nbsp;:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Find\\ the\\ moment\\ of\\ the\\ force\\ 3\\overrightarrow{i}+4\\overrightarrow{j}+5\\overrightarrow{k}\\ acting\\ through\\ the\\ point\\ \\overrightarrow{i}-2\\overrightarrow{j}+3\\overrightarrow{k} about\\ the\\ point\\  4\\overrightarrow{i}- 3\\overrightarrow{j}+\\overrightarrow{k}.\\]<\/div>\n\n\n\n<p>Soln:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 3\\overrightarrow{i} +4\\overrightarrow{j} + 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= \\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 4\\overrightarrow{i} -3\\overrightarrow{j} +\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}= \\overrightarrow{AP} = \\overrightarrow{OP}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}- (4\\overrightarrow{i} -3\\overrightarrow{j} +\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}- 4\\overrightarrow{i} +3\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}=  -3\\overrightarrow{i} + \\overrightarrow{j} + 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Moment = \\overrightarrow{r}\u00d7 \\overrightarrow{F}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n-3 &amp; 1 &amp; 2\\\\\n3 &amp; 4 &amp; 5\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 5 &#8211; 8) -\\overrightarrow{j}(-15 &#8211; 6)+\\overrightarrow{k}(-12 &#8211; 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}\u00d7 \\overrightarrow{F}= -3\\overrightarrow{i}+ 21\\overrightarrow{j}-15\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Magnitude\\ of \\ Moment = |\\overrightarrow{r} \u00d7 \\overrightarrow{F}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\sqrt{(-3)^2 + (21)^2 + (-15)^2 }=\\sqrt{(9 + 441+ 225 }=\\sqrt{675}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Magnitude\\ of \\ Moment = \\sqrt{675}\\ units\\]<\/div>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/shareasale.com\/r.cfm?b=1752996&amp;u=2863840&amp;m=108359&amp;urllink=&amp;afftrack=\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/static.shareasale.com\/image\/108359\/OrganicCottonBathRobe.jpg?ssl=1\" border=\"0\"><\/a>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/www.amazon.in\/gp\/product\/B095KQ1G3X\/ref=as_li_tl?ie=UTF8&amp;camp=3638&amp;creative=24630&amp;creativeASIN=B095KQ1G3X&amp;linkCode=as2&amp;tag=yanamtakshash-21&amp;linkId=b8a3a9d4e80df30f57e22533549573b8\" rel=\"noopener\"><img decoding=\"async\" border=\"0\" src=\"\/\/ws-in.amazon-adsystem.com\/widgets\/q?_encoding=UTF8&amp;MarketPlace=IN&amp;ASIN=B095KQ1G3X&amp;ServiceVersion=20070822&amp;ID=AsinImage&amp;WS=1&amp;Format=_SL160_&amp;tag=yanamtakshash-21\"><\/a>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/www.amazon.in\/gp\/product\/B083VR318D\/ref=as_li_tl?ie=UTF8&amp;camp=3638&amp;creative=24630&amp;creativeASIN=B083VR318D&amp;linkCode=as2&amp;tag=yanamtakshash-21&amp;linkId=2fb1c2aa5ef2d8bfb229e82c77ae4b17\" rel=\"noopener\"><img decoding=\"async\" border=\"0\" src=\"\/\/ws-in.amazon-adsystem.com\/widgets\/q?_encoding=UTF8&amp;MarketPlace=IN&amp;ASIN=B083VR318D&amp;ServiceVersion=20070822&amp;ID=AsinImage&amp;WS=1&amp;Format=_SL160_&amp;tag=yanamtakshash-21\"><\/a>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Application of Scalar Product Work done PART &#8211; B Example &nbsp;: Soln: = 3 ( 2 ) + 5 ( &#8211; 1 ) + 7 ( 1 ) =&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 5&nbsp; +&nbsp; 7 Work done = 8 units Example &nbsp;: Soln: = 4 ( 2 ) + 9 ( &#8211; 7 ) + 4 [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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